In looking at the data and/or the scatter plot, not all of the 5-year growths are the same. Therefore, there is some variation in the response variable. The hope is that the least-squares regression line will fit between the data points in a manner that will “explain” quite a bit of that variation. The closer the data points are to the regression line, the higher proportion of the variation in the response variable that’s explained by the regression line. Categorical variables are also useful in predicting outcomes. Here we consider a categorical predictor with two levels (recall that a level is the same as a category).

For example, say we have a list of how many topics future engineers here at freeCodeCamp can solve if they invest 1, 2, or 3 hours continuously. Then we can predict how many topics will be covered after 4 hours of continuous study even without that data being available to us. Using the R output, write the equation of the least-squares regression line. The slope indicates that, on average, new games sell for about $10.90 more than used games. Example 7.22 Interpret the two parameters estimated in the model for the price of Mario Kart in eBay auctions.

## Error

The better the line fits the data, the smaller the residuals (on average). In other words, how do we determine values of the intercept and slope for our regression line? Intuitively, if we were to manually fit a line to our data, we would try to find a line that minimizes the model errors, overall.

Let's assume that an analyst wishes to test the relationship between a company’s stock returns, and the returns of the index for which the stock is a component. In this example, the analyst seeks to test the dependence of the stock returns on the index returns. In other words, for any other line other than the LSRL, the sum of the residuals squared will be greater. Imagine you have a scatterplot full of points, and you want to draw the line which will best fit your data.

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Investors and analysts can use the least square method by analyzing past performance and making predictions about future trends in the economy and stock markets. One of the main benefits of using this method is that it is easy to apply and understand. That's because it only uses two variables (one that is shown along the x-axis and the other on the y-axis) while highlighting the best relationship between them. A least squares regression line can be used to predict the value of y if the corresponding x value is given. It implies a cause-and-effect relationship between x and y and ensures that the predictions of y outside the range of the values of x are valid. It can only be determined if a good linear relationship exists between x and y.

## What is the Least Squares Regression method and why use it?

Although it may be easy to apply and understand, it only relies on two variables so it doesn't account for any outliers. That's why it's best used in conjunction with other analytical tools to get more reliable results. The least squares method is a form of regression analysis that provides the overall rationale for the placement of the line of best fit among the data points being studied. It begins with a set of data points using two variables, which are plotted on a graph along the x- and y-axis. Traders and analysts can use this as a tool to pinpoint bullish and bearish trends in the market along with potential trading opportunities.

It is possible to find the (coefficients of the) LSRL using the above information, but it is often more convenient to use a calculator or other electronic tool. To achieve this, all of the returns are plotted on a chart. The index returns are then designated as the independent variable, and the stock returns are the dependent variable. The line of best fit provides the analyst with coefficients explaining the level of dependence.

A scatterplot shows points that are all very close to theregression line.D. In the past two lessons, we’ve mentioned fitting a line between the points. In this lesson, we’ll discuss how to best “fit” a line between the points if the relationship between the response and explanatory variable is linear. This “best-fitting” line is called the least-squares regression line and can be described by an equation.

## A Quadratic Equation

Let’s lock this line in place, and attach springs between the data points and the line. The least-squares regression method finds the a and b making the sum of squares error, E, as small as possible. Try the following example problems for analyzing data sets using the least-squares regression method.

Our fitted regression line enables us to predict the response, Y, for a given value of X. She may use it as an estimate, though some qualifiers on this approach are important. First, the data all come from one freshman class, and the way aid is determined by the university may change from year to year. While the linear equation is good at capturing the trend in the data, no individual student's aid will be perfectly predicted. This method, the method of least squares, finds values of the intercept and slope coefficient that minimize the sum of the squared errors.

## What is Least-Squares Regression?

The model predicts this student will have -$18,800 in aid (!). Elmhurst College cannot (or at least does not) require any students to pay extra on top of tuition to attend. The trend appears to be linear, the data fall around the line with no obvious outliers, the variance is roughly constant. So, when we square each of those errors and add them all up, the total is as small as possible. X- is the mean of all the x-values, y- is the mean of all the y-values, and n is the number of pairs in the data set.

Rather, we will rely on obtaining and interpreting output from R to determine the values of the slope and y-intercept. Even so, the formulas are included as an “appendix” to this lesson so that you are aware of how R determines these values. The line that minimizes the vertical distance between the points and the line that fits them (aka the least-squares regression line). We will help Fred fit a linear equation, a quadratic equation, and an exponential equation to his data.

This best line is the Least Squares Regression Line (abbreviated as LSRL). Someone needs to remind Fred, the error depends on the equation choice and the data scatter. In the method, N is the number of data points, while x and y are the coordinates of the data points. For categorical predictors with a fitted least squares regression line just two levels, the linearity assumption will always be satis ed. However, we must evaluate whether the residuals in each group are approximately normal and have approximately equal variance. As can be seen in Figure 7.17, both of these conditions are reasonably satis ed by the auction data.

## Lesson Summary

There isn't much to be said about the code here since it's all the theory that we've been through earlier. We loop through the values to get sums, averages, and all the other values we need to obtain the coefficient (a) and the slope (b). We can create our project where we input the X and Y values, it draws a graph with those points, and applies the linear regression formula. The primary disadvantage of the least square method lies in the data used. It can only highlight the relationship between two variables.

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It's a powerful formula and if you build any project using it I would love to see it. Regardless, predicting the future is a fun concept even if, in reality, the most we can hope to predict is an approximation based on past data points. We have to grab our instance of the chart and call update so we see the new values being taken into account. We have the pairs and line in the current variable so we use them in the next step to update our chart.

## Using R2 to describe the strength of a fit

But, when we fit a line through data, some of the errors will be positive and some will be negative. Least-squares regression is a method to find the least-squares regression line (otherwise known as the line of best fit or the trendline) or the curve of best fit for a set of data. That line minimizes the sum of the residuals, or errors, squared. Because two points determine a line, the least-squares regression line for only two data points would pass through both points, and so the error would be zero.

- Squaring eliminates the minus signs, so no cancellation can occur.
- Least-squares regression is used to determine the line or curve of best fit.
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- One of the main benefits of using this method is that it is easy to apply and understand.
- Least-squares regression is also used to illustrate a trend and to predict or estimate a data value.

The truth is almost always much more complex than our simple line. For example, we do not know how the data outside of our limited window will behave. Be cautious about applying regression to data collected sequentially in what is called a time series. Such data may have an underlying structure that should be considered in a model and analysis.